Specialized to the case when \({C}=\mathit {Set}\) , we see that this data
amounts to an action of the simplicial group \(G\)  on \(P\)  over
\(M\)  such that \(G_n\)  acts freely and transitively on \(P_n\)  for
each \(n\) , and also the existence of
a section \(\sigma \colon \mathrm {Dec}_0 M\rightarrow \mathrm {Dec}_0 P\) . Such a section
corresponds exactly to the notion of pseudo-cross section
(see for example [1]} Definition 18.5).
